So is the *number* of solutions to ? Are and distinct? For example for , are 6=4+2 and 6=2+4 two separate solutions, so , or is ?
Let an denote the solutions of n = x1 + x2 where x1, x2 are natural numbers and x2 is an even number.
I need to show that the generating function of an is of the form:
(1 + x + x^2 + x^3 + ...)(1 + x^2 + x^4 + x^6 + ...) = 1/[(1-x)(1-x^2)]
I ultimately want to find a closed formula for an. However I would like to prove the above first before I go about doing that. Any ideas would be great. Thanks!
You just have to think about the coefficient of in the product
In order to get an , you need one term from
which can be any power of x, say
and one term from
which can be any even power of x, say .
Then you must have ,
The number of solutions to this equation is the coefficient of in the product.